There are many sources of cut green branches, if not in your own back yard, then at your local city, county, state or national parks departments, college campuses, arboretums, etc.  Get to know the tree trimmers there and find out what they do with their trimmings.  Some may feed them directly into a wood chipper.  Go there and rescue a beautiful branch before she's turned into mulch. 

If a branch is to be trimmed from a tree in your yard or property, so much the better. 
Working with a branch from a tree that you know can deepen the connection you feel with the wood in the making of a flute.  And it's a wonderful experience to play a finished branch flute under the tree that gave her birth.  If you prune your own branches you have the opportunity to make the cuts with the length of the flute in mind.  If gathering them from someone else's branch pile, you can cut them to size with a good pruning saw so that you only take what you need and so that the branches fit in your car or truck.  Assuming that you're already familiar with bore length/diameter ratio, first visualize the flute in a section of branch and then make your cuts, if you can, a few inches longer at each end.  This will allow for any end checking that might happen before you get the green branch back to the shop.  Once there, you can cut it again, closer to your intended length, saving a section to use for the bird (block, fetish) and then seal the ends.

The Honorable Harvest
Traditional native cultures have gathered green tree branches for many uses, including bows, arrows, lodge poles, basket making, as well as for flutes.  This is done, though, with a deep understanding and kinship with the natural community from which one is taking, with respect for the principles of the Honorable Harvest:  

~ Ask permission of the ones whose lives you seek. Abide by the answer.

~ Never take the first. Never take the last.

~ Harvest in a way that minimizes harm.

~ Take only what you need and leave some for others.

~ Use everything that you take.

~ Take only that which is given to you.

~ Share it, as the Earth has shared with you.

~ Be grateful.

~ Reciprocate the gift.

Pruning a green tree branch, if done properly, can increase the health and longevity of the tree,
as the elements are not always the best stewards.  Good pruning is both a science and an art, the fundamentals of which, if you're capable of making a flute, you're certainly capable of practicing.  Some folks might think that downed wood is free for the taking, since it has died.  It is, however, no more a gift and no more free or ethical to gather than a green branch pruned directly from the tree.  That "dead" branch is still a very important part of life, playing essential roles in the ecosystem in which it is found.Dying Tree  She may be a source of food and housing for countless insects.  She may be providing shelter for small creatures such as wood frogs, salamanders, lizards, and snakes.  She may be serving as a protective nursery for sprouting plants.  She may be a place for turtles and other reptiles to sun themselves.  She may be recycling her nutrients back into the soil as well as helping to stabilize that soil, reducing erosion by wind, rain and melting snow.  In streams and lakes, downed wood, or driftwood, may be providing shelter for fish, the insects and algae living in that wood providing food for many water dwelling creatures.
So whether one is gathering downed or green wood, it aught to be practiced 
with awareness and sensitivity to the whole environment from which the wood is taken and with knowledge of the ecological laws and legal regulations governing the area the wood is in.  An honorable harvest includes reciprocation, so you'll want to give something in exchange, out of gratitude to the tree or spirit of the place from which you've gathered--some cornmeal or tobacco or a sip of water or a prayer.  All life is sacred.  And as John Muir wrote, “When we try to pick out anything by itself, we find it hitched to everything else in the universe.”  With a good heart then, and while leaving as light a footprint as you can, gather ye branches while ye may!

A very good article by Robin Wall Kimmerer, of the Potowatomi Nation, concerning the Honorable Harvest.

A link to a good manual on tree pruning.

For more information on the role of downed wood in ecosystems, please reference this brochure: